The ball and socket joint at the end of a rocker arm has a h

The ball and socket joint at the end of a rocker arm has a hardened-steel (E = 207 GPa, v = 0.3) spherical surface 10 mm in diameter fitting in a hard-bronze (E = 110 GPa, v = 0.33) bearing alloy spherical seat 10.1 mm in diameter. What maximum contact stress (normal and shear) will result from a load of 2000 N? Also, specify the location of these stresses.

Solution

Ball and Socket joint of rocker arm:

Given:

For Spherical surface 1:-

E1 = 207 GPa    and n1 = 0.3 and d1 = 10mm

For Spherical surface 2 (cup shape):

E2 =110 GPa      and n2 = 0.33 and d2 = 10.1mm

Where E is young’s modulus and u is Poisson’s ratio.

When two spherical surfaces are pressed with a force F (=2000N) the contact surface becomes circular with radius r given by:

Consider d2 as negative since the sphere is in contact with internal phase of second sphere.

a = 2.115mm

diameter of the circle formed at at the contact surface due the load 2000n is 4.230 mm.

the maximum pressure occur at centre of contact area is given by

Pmax = 213.390MPa

The pressure distribution is hemispherical with in each contact area

The maximum stresses occur on the axis passing through the direction of load (let it be z axis) and they are called principal stresses and given by:

s1 = s2 = -Pmax

(where z is the distance of the point on the on z axis from contact surface)

s3=

When z=0

s1=s2 = 0.8 * 213.390 Mpa =170.712 MPa. For sphere 1.

s1= 177.1137 Mpa. For sphere 2.

s3 = -213.390Mpa.

When z= a

s1= 6.295 MPa for sphere 1

s1=7.671 Mpa

s3= -106.695Mpa.

The derivative for maxima of the above equation for s gives the location for maximum Principal stress.